Chiziqli funksiya

⭐ — imtihonda eng ko'p ishlatiladigan formulalar. · 🟢 — masalani tezlashtiradi · 🟡 — ilg'or daraja · 🔵 — chuqur o'rganish (olimpiada darajasi)

Gradient (burchak koeffitsienti)

• ⭐ Gradient: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$, shuningdek $m = \tan\theta$ ($\theta$ — chiziq bilan $x$ o'qi orasidagi burchak)
• Gorizontal chiziq gradienti $m = 0$; vertikal chiziq gradienti aniqlanmagan.
• ⭐ Parallel chiziqlar: $m_1 = m_2$
• ⭐ Perpendikulyar chiziqlar: $m_1 m_2 = -1$, ya'ni $m_2 = -\dfrac{1}{m_1}$

To'g'ri chiziq tenglamasining shakllari

• O'rta perpendikulyar kesmaning o'rta nuqtasidan o'tadi, gradienti $-\dfrac{1}{m}$.
• Ikki chiziqning kesishish nuqtasi — ularning tenglamalarini sistema qilib yechish natijasi. Parallel chiziqlar kesishmaydi (sistemaning yechimi yo'q).

Boshqa foydali formulalar

• Uchburchakning og'irlik markazi (centroid): $\left(\dfrac{x_1 + x_2 + x_3}{3},\ \dfrac{y_1 + y_2 + y_3}{3}\right)$
• Kesmalar shakli: $\dfrac{x}{a} + \dfrac{y}{b} = 1$. Bu chiziq $(a, 0)$ va $(0, b)$ nuqtalardan o'tadi.

🟢 $ax + by + c = 0$ chizig'i bilan ishlash

• Gradienti: $m = -\dfrac{a}{b}$, $y$ o'qini kesish nuqtasi: $-\dfrac{c}{b}$
• Unga parallel chiziq: $ax + by + k = 0$
• Unga perpendikulyar chiziq: $bx - ay + k = 0$

($k$ ni berilgan nuqtani qo'yib topasiz.)

🟢 Burchak orqali chiziq

$x$ o'qi bilan $\theta$ burchak hosil qiluvchi va $(x_1, y_1)$ dan o'tuvchi chiziq: $\;y - y_1 = \tan\theta\,(x - x_1)$

🟢 Ikki chiziq orasidagi burchak

$\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$

$1 + m_1 m_2 = 0$ bo'lsa, chiziqlar perpendikulyar.

🟢 Nuqtadan chiziqqacha va parallel chiziqlar orasidagi masofa

$d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}} \qquad d_{\parallel} = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}$

Misol: $(1, 2)$ dan $3x + 4y - 1 = 0$ gacha: $d = \dfrac{|3 + 8 - 1|}{5} = 2$

🟢 O'rta perpendikulyarning tayyor tenglamasi

$A(x_1, y_1)$ va $B(x_2, y_2)$ dan teng uzoqlikdagi nuqtalar to'plami:

$2(x_2 - x_1)\,x + 2(y_2 - y_1)\,y = x_2^2 + y_2^2 - x_1^2 - y_1^2$

🟢 Perpendikulyar asosi va nuqtaning aksi

$P(x_0, y_0)$ va $ax + by + c = 0$ uchun $t = \dfrac{ax_0 + by_0 + c}{a^2 + b^2}$:

• Perpendikulyar asosi: $H = (x_0 - a\,t,\ \ y_0 - b\,t)$
• Chiziqqa nisbatan aksi: $P' = (x_0 - 2a\,t,\ \ y_0 - 2b\,t)$

🔵 Ikki chiziq kesishish nuqtasidan o'tuvchi chiziqlar oilasi

$(a_1x + b_1y + c_1) + \lambda\,(a_2x + b_2y + c_2) = 0$

Kesishish nuqtasini topmasdan, $\lambda$ ni qo'shimcha shartdan topasiz.

🔵 Burchak bissektrisalari

$\frac{a_1x + b_1y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm\,\frac{a_2x + b_2y + c_2}{\sqrt{a_2^2 + b_2^2}}$

🔵 Uch chiziqning bir nuqtada kesishish sharti

$\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0$